Miniversal deformations of dialgebras

1 2 3 MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS ALICE FIALOWSKI AND ANIT A MAJUMDAR Abstra ct. W e develop the theory of versal deformations of dialgebras and describe a method for constructing a miniversal deformation of a dialgebra. 1. Introduction The notion of Leibn iz algebras and d ialgebras wa s disco v ered b y J.-L. Lo day while studying p erio dicit y phen omena in algebraic K-theory [10]. Leibniz algebras are a non-comm utativ e v ariation of Lie algebras and dialgebras are a v ariation of asso ciativ e algebras. Recall that any asso ciativ e algebra give s r ise to a Lie algebra b y [ x, y ] = xy − y x . The n otion of dialgebras w as in v en ted in ord er to build analogue of the couple Lie algebras ↔ asso ciativ e a lgebras , where L ie algebras are replaced b y Leibniz algebras. Shortly , dialgebra is to Leibniz algebra, what asso ciativ e algebra is to Lie algebra. A (co)homo logy theory asso ciated to d ialgebras was d ev elop ed by J.-L. Lo da y , called the dialgebra cohomology where planar binary trees play a crucial role in the construction. Dialgebra cohomology with coefficient s was studied by A. F rab etti [5, 6] and deformations of dialgebras w ere dev elop ed in [11]. In the present pap er, we deve lop a deform ation theory of dialge- bras o v er a comm u tativ e un ital algebra base, follo wing [2], and sho w that dialgebra cohomology is a natural candidate for the cohomology con trolling the deformations. W e work out a constru ction of a v ersal deformation for dialgebras, follo win g [3]. The p ap er is organized as follo w s. In Section 2, we recall some facts on dialgebra and its cohomology . In Section 3, we introdu ce the definitions of deformations of dialgebras o v er a commuta tiv e, unital algebra b ase. In Section 4, w e pro duce an example of an infin itesimal deformation of a dialgebra D o ver a field K , denoted b y η D , and also sh o w that this deformation is co-universal in the sense that, giv en an y infin itesimal d eformation λ of a dialgebra D with a fin ite d imensional base A , there exists a unique homomorphism φ : K ⊕ H Y 2 ( D , D ) ′ − → A , where H Y 2 ( D , D ) denotes the tw o dimensional cohomology of D w ith co efficients in itself, su c h that λ is equiv alen t to the pu s h-out φ ∗ η D . Section 5 comprises resu lts of Harrison cohomology of a comm utativ e unital algebra A with co efficient s in a A -mo dule M , which hav e b een used in the pap er. In Section 6 we int ro duce obstructions to extending a deformation o ver a base A , to a deformation o ver a base B , w here there exists an extension 1 AMS Mathematics Sub ject Classification : 54H20, 57S25. 2 The first auth or w as sup p orted b y gran ts OTKA T043641 and T043034. 3 The second author is supp orted by NBHM Post-doctoral fello wship. Key wor ds and phr ases. Dialgebras, cohomology , versa l, miniv ersal, deformations . 1 2 ALICE FIALOWS KI AND ANIT A MAJUM DAR 0 → K i → B p → A → 0 of A . W e sho w that an obstruction is a cohomology class, v anishing of whic h is a necessary and sufficien t cond ition for the given deformation of D o v er base A to b e extended to a deformation of D o v er base B . In Section 7 w e d iscuss extendible deformations. Let λ b e the d eformation of D ov er A which is extendib le. W e state that the t w o dimensional cohomology group H Y 2 ( D , D ) op erates transitiv ely on th e set of equ iv alence classes of d eformations µ of D with base B suc h that p ∗ µ = λ. W e also state that the group of automorphisms of the extension 0 → K i → B p → A → 0 op erates on the set of equiv alence classes of deformations µ suc h that p ∗ µ = λ . These t wo actions are related b y a map calle d the differen tial dλ : T A → H Y 2 ( D , D ) , where T A denotes the tangen t space of A . In the last section, we presen t a construction of a min iv er s al deformation of a dialgebra D . 2. Dialgebra an d its Cohomology Throughto ut this pap er, K will denote the groun d field of characte ristic zero. All tensor pro d ucts shall b e o ve r K unless sp ecified. In this section, we recall the defini- tion of a dialgebra and the constru ction of the dialgebra co c h ain complex. S ince w e are interested in co efficien ts in th e dialgebra itself, we shall restrict our definition to the same. Definition 2.1. A dialgebra D o ver K is a ve ctor space o v er K along with t w o K - linear maps ⊣ : D ⊗ D − → D called left and ⊢ : D ⊗ D − → D called right satisfying the follo win g axioms : x ⊣ ( y ⊣ z ) 1 = ( x ⊣ y ) ⊣ z 2 = x ⊣ ( y ⊢ z ) ( x ⊢ y ) ⊣ z 3 = x ⊢ ( y ⊣ z ) ( x ⊣ y ) ⊢ z 4 = x ⊢ ( y ⊢ z ) 5 = ( x ⊢ y ) ⊢ z (2.1.1) for all x, y , z ∈ D . Apart fr om the kno wn algebraic examples of dialgebras, [10], we cite an in teresting example of a family of dialgebras in the con text of functional analysis, [1]. Example 2.2. Let H b e a Hilb ert space and e ∈ H with || e || = 1. Define t wo linear op erations ⊣ and ⊢ by a ⊣ b = h b, e i a, a ⊢ b = h a, e i b, for a, b ∈ H . Then ( H , ⊣ , ⊢ ) is a dialgebra, more pr ecisely , a n ormed dialgebra [1]. A morphism φ : D − → D ′ b et w een tw o dialgebras is a K-linear map such that φ ( x ⊣ y ) = φ ( x ) ⊣ φ ( y ) and φ ( x ⊢ y ) = φ ( x ) ⊢ φ ( y ) . A planar binary tree with n v ertices (in short, n -tree) is a planar tree with ( n + 1) lea ve s, one ro ot and eac h v ertex triv alent. Let Y n denote the set of all n -trees. Let Y 0 b e the singleton set consisting of a r o ot only . The n -trees for 0 ≤ n ≤ 3 are give n b y the follo w ing diagrams: MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 3 = { , , , , } . Y 3 } , Y 1 = { } , Y 2 , } , { = { = 0 Y F or any y ∈ Y n , the ( n + 1) lea v es are lab elled by { 0 , 1 , . . . , n } from left to r igh t and the v ertices are lab elled { 1 , 2 , . . . , n } so that the i th v ertex is b et ween the leav es ( i − 1) and i . T he on ly elemen t | of Y 0 is d enoted by [0] and the only elemen t of Y 1 is denoted b y [1]. Th e grafting of a p -tree y 1 and a q -tree y 2 is a ( p + q + 1)-tree denoted b y y 1 ∨ y 2 whic h is ob tained b y joining the ro ots of y 1 and y 2 and creating a new ro ot from th at v ertex. Th is is denoted by [ y 1 p + q + 1 y 2 ] with the conv en tion that all zeros are deleted except for the elemen t in Y 0 . With this n otation, the trees pictured ab o v e from left to righ t are [0] , [1] , [12] , [21 ] , [123] , [213] , [131] , [312] , [321]. F or an y i , 0 ≤ i ≤ n , there is a map, called the face map, d i : Y n − → Y n − 1 , y 7→ d i y where d i y is obtained from y by deleting the i th leaf. The face maps satisfy the relations d i d j = d j − 1 d i , for all i < j . Let D b e a dialgebra o v er a field K . Th e coc hain complex C Y ∗ ( D , D ) whic h defines the dialgebra cohomology H Y ∗ ( D , D ) is defined as follo ws . F or an y n ≥ 0, let K [ Y n ] denote the K -v ector space spann ed by Y n and C Y n ( D , D ) := Hom K ( K [ Y n ] ⊗ D ⊗ n , D ) b e the mo du le of n -cochains of D w ith co efficients in D . The cob oundary op erator δ : C Y n ( D , D ) − → C Y n +1 ( D , D ) is d efi ned as the K -linear map δ = P n +1 i =0 ( − 1) i δ i , where ( δ i f )( y ; a 1 , a 2 , . . . , a n +1 ) =    a 1 ◦ y 0 f ( d 0 y ; a 2 , . . . , a n +1 ) , i = 0 f ( d i y ; a 1 , . . . , a i ◦ y i a i +1 , . . . , a n +1 ) , 1 ≤ i ≤ n f ( d n +1 y ; a 1 , . . . , a n ) ◦ y n +1 a n +1 , i = n + 1 for an y y ∈ Y n +1 ; a 1 , . . . , a n +1 ∈ D an d f : K [ Y n ] ⊗ D ⊗ n − → D . Here, for an y i , 0 ≤ i ≤ n + 1. Th e maps ◦ i : Y n +1 − → {⊣ , ⊢} , are d efined by ◦ 0 ( y ) = ◦ y 0 :=  ⊣ if y is of the form | ∨ y 1 , for s ome n -tree y 1 ⊢ otherwise ◦ i ( y ) = ◦ y i :=  ⊣ if the i th leaf of y is orien ted lik e ‘ \ ’ ⊢ if the i th leaf of y is orien ted lik e ‘ / ’ for 1 ≤ i ≤ n and ◦ n +1 ( y ) = ◦ y n +1 :=  ⊢ if y is of the form y 1 ∨ | , for some n -tree y 1 ⊣ otherwise , where the symb ol ‘ ∨ ′ stands for grafting of trees [10]. 4 ALICE FIALOWS KI AND ANIT A MAJUM DAR There exists a pre-Lie algebra structur e on C Y ∗ ( D , D ), [11, 12], the pre-Lie pro duct b eing denoted by ◦ : C Y n ( D , D ) ⊗ C Y m ( D , D ) − → C Y n + m − 1 . Also, if w e mo dify the cob oundary map δ b y a sign, sa y dx = ( − 1) | x | δ ( x ), and d efine a brac ket pro du ct on C Y ∗ ( D , D ) b y [ x, y ] = x ◦ y − ( − 1) | x || y | y ◦ x , whic h is the comm u tator of th e pre-Lie pro d uct, then ( C Y ∗ ( D , D ) , d ) forms a differen tial graded Lie algebra, [12], where | x | = deg x − 1. 3. Def orma tions of Dialgebras Let D b e a dialgebra o v er K and let A b e a comm u tativ e u nital algebra ov er K with a fi xed augmentati on ǫ : A − → K with ǫ (1) = 1 . Let m = k er ǫ. W e assum e dim ( m k / m k +1 ) < ∞ , f or all k ≥ 1. Definition 3.1. A deformation λ of D with base ( A, m ) is a dialgebra structure on the tensor pro du ct A ⊗ K D with the pro d ucts ⊣ λ and ⊢ λ b eing A -linear (or simply , an A -dialgebra s tr ucture) such th at ǫ ⊗ id : A ⊗ K D − → K ⊗ D ∼ = D is a A -linear dialgebra morph ism. The left action of A on K ⊗ D is giv en by the augmentat ion map. W e n ote that for x 1 , x 2 ∈ D , and a, b ∈ A , a ⊗ x 1 ∗ λ b ⊗ x 2 = ab (1 ⊗ x 1 ∗ 1 ⊗ x 2 ) , b y A-linearit y of the pro ducts, wh ere ∗ = {⊣ , ⊢} . Also, since ǫ ⊗ id : A ⊗ D − → K ⊗ D is a A -linear dialgebra homomorphism, ( ǫ ⊗ id) { 1 ⊗ x 1 ∗ λ 1 ⊗ x 2 } = ( ǫ ⊗ id )(1 ⊗ x 1 ) ∗ ( ǫ ⊗ id)(1 ⊗ x 2 ) = (1 ⊗ x 1 ) ∗ (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ∗ x 2 ) = ( ǫ ⊗ id)(1 ⊗ ( x 1 ∗ x 2 )) . So, (1 ⊗ x 1 ) ∗ λ (1 ⊗ x 2 ) − 1 ⊗ ( x 1 ∗ x 2 ) ∈ k er( ǫ ⊗ id). Hence (1 ⊗ x 1 ) ∗ λ (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ∗ x 2 ) + P i m i ⊗ d i , wh ere m i ∈ ke r ǫ = m and d i ∈ D and P i m i ⊗ d i is a finite sum. Definition 3.2. T wo deformations of D with the same base A are called equiv alen t if there exists a A -linear dialgebra isomorp hism b et w een the t wo copies of A ⊗ D with the t w o dialgebra stru ctures, compatible with ǫ ⊗ id . A d eformation of D with b ase A is calle d local if the algebra A is lo cal, and will b e calle d infi n itesimal if, in addition, m 2 = 0, w here m is the maximal ideal of A . Definition 3.3. Let A b e a complete lo cal algebra, th at is, A = ← − − lim n →∞ ( A/ m n ), m denoting the maximal ideal in A . A formal deformation of D with base A is a A - dialgebra stru cture on the complete d tensor pr o duct A b ⊗ D = ← − − lim n →∞ ( A/ m n ) ⊗ D ), suc h that ǫ b ⊗ id : A b ⊗ D − → K ⊗ D = D is a A -linear dialgebra m orphism. MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 5 Tw o formal deformations of a dialgebra D with the same base A are called equiv alen t if there exists a dialgebra isomorphism b etw een the tw o copies of A b ⊗ D with the t wo dialgebra structur es compatible with ǫ b ⊗ id . Example 3.4. If A = K [[ t ]] then a formal deform ation of D with b ase A is the same as a formal one-parameter deformation of D , [11]. Let A ′ b e a commutati v e algebra with iden tity , with a fi xed augmen tation ǫ ′ : A ′ − → K and let φ : A − → A ′ b e an algebra homomorphism, with φ (1) = 1 and ǫ ′ ◦ φ = ǫ. Then w e can constru ct a d eformation of D with base A ′ in the follo wing w ay . Definition 3.5. Let λ b e a deformation of the d ialgebra D with base ( A, m ). The push - out φ ∗ λ is th e deformation of D with b ase ( A ′ , m ′ = k er ǫ ′ ), w hic h is th e dial- gebra structure give n b y a ′ 1 ⊗ A ( a 1 ⊗ x 1 ) ⊣ φ ∗ λ a ′ 2 ⊗ A ( a 2 ⊗ x 2 ) = a ′ 1 a ′ 2 ⊗ A ( a 1 ⊗ x 1 ⊣ λ a 2 ⊗ x 2 ) a ′ 1 ⊗ A ( a 1 ⊗ x 1 ) ⊢ φ ∗ λ a ′ 2 ⊗ A ( a 2 ⊗ x 2 ) = a ′ 1 a ′ 2 ⊗ A ( a 1 ⊗ x 1 ⊢ λ a 2 ⊗ x 2 ) , where a 1 , a 2 ∈ A ′ , a 1 , a 2 ∈ A and l 1 , l 2 ∈ D . Here we mak e use of the fact that A ′ ⊗ D = ( A ′ ⊗ A A ) ⊗ D = A ′ ⊗ A ( A ⊗ D ) , wh ere A ′ is regarded as an A -mo du le by the structure a ′ a = a ′ φ ( a ) . Similarly , one can define the push - out of formal d eformations. Remark 3.6. W e note that if the dialgebra stru cture λ on A ⊗ D is given b y (1 ⊗ x 1 ) ∗ λ (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ∗ x 2 ) + n X i =1 m i ⊗ d i ; m i ∈ m , d i ∈ D , then the dialgebra structure φ ∗ λ on A ′ ⊗ D is giv en by (1 ⊗ x 1 ) ∗ φ ∗ λ (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ∗ x 2 ) + n X i =1 φ ( m i ) ⊗ d i . 4. Universal In finitesimal an d Miniver sal Deforma tions of Dialgebras In [3], the authors ha v e pro du ced a fun dament al example of an infinitesimal deforma- tion of Lie algebras. Here w e p ro du ce an example of an infin itesimal d eform ation of dialgebras, w hic h is obtained from the aforesaid example, with sligh t mo difications. Supp ose d im H Y 2 ( D , D ) < ∞ . This is, in particular, true if dim D < ∞ . Consider the base of the deformation to b e A = K ⊕ H Y 2 ( D , D ) ′ , with ′ denoting the linear dual. Here, A is local with the maximal ideal m = H Y 2 ( D , D ) ′ , and m 2 = 0 . Let µ : H Y 2 ( D , D ) − → C Y 2 ( D , D ) = Hom ( K [ Y 2 ] ⊗ D ⊗ 2 , D ) 6 ALICE FIALOWS KI AND ANIT A MAJUM DAR whic h tak es a cohomology class into a cocycle represen ting the class. Define a dial- gebra structure on A ⊗ D = ( K ⊕ H Y 2 ( D , D ) ′ ) ⊗ D = ( K ⊗ D ) ⊕ ( H Y 2 ( D , D ) ′ ⊗ D ) = D ⊕ ( H Y 2 ( D , D ) ′ ⊗ D ) = D ⊕ Hom ( H Y 2 ( D , D ) , D ) b y ( x 1 , φ 1 ) ⊣ ( x 2 , φ 2 ) = ( x 1 ⊣ x 2 , ψ ℓ ) ( x 1 , φ 1 ) ⊢ ( x 2 , φ 2 ) = ( x 1 ⊢ x 2 , ψ r ) where ψ ℓ ( α ) = µ ( α )([21]; x 1 , x 2 ) + φ 1 ( α ) ⊣ x 2 + x 1 ⊣ φ 2 ( α ) ψ r ( α ) = µ ( α )([12] ; x 1 , x 2 ) + φ 1 ( α ) ⊢ x 2 + x 1 ⊢ φ 2 ( α ) , for α ∈ H Y 2 ( D , D ). Using the d ialgebra structure of D an d the fact that µ ( α ) is a 2-co cycle of D , one can c hec k that the ⊣ and ⊢ pro ducts d efined this wa y satisfy th e dialgebra axioms. It is to b e noted that this deformation do es not d ep end on the c hoice of µ , upto an isomorphism. Let µ ′ : H Y 2 ( D , D ) − → C Y 2 ( D , D ) b e another c h oice of µ. Define a homomorph ism ν : H Y 2 ( D , D ) − → C Y 1 ( D , D ) ∼ = Hom( D , D ) b y µ ′ ( α ) − µ ( α ) = δ ν ( α ) , f or all α ∈ H Y 2 ( D , D ). W e define a linear automorphism ρ of the space A ⊗ D = D ⊕ Hom( H Y 2 ( D , D ) , D ) by ρ ( x, φ ) = ( x, ψ ) where ψ ( α ) = φ ( α ) + ν ( α )( x ) . It is str aightforw ard to c hec k that ρ defin es a dialgebra isomorphism b et w een the tw o dialgebra stru ctures indu ced b y µ and µ ′ resp ectiv ely . W e denote the infinitesimal d eformation of D as constructed ab o ve b y η D . Belo w we will sho w th e couniversalit y of η D in the class of infinitesimal deformations: Let λ b e an infinitesimal deformation of the dialgebra D , w ith a finite dimensional lo cal algebra base A , with m 2 = 0, where m is the maximal id eal of A . Let ξ ∈ m ′ = Hom K ( m , K ) . This is equiv alent to ξ ∈ Hom K ( A, K ) with ξ (1) = 0 . F or x 1 , x 2 ∈ D , let u s define a 2-coc hain as follo ws: α λ,ξ ([21]; x 1 , x 2 ) = ( ξ ⊗ id)((1 ⊗ x 1 ) ⊣ λ (1 ⊗ x 2 )) and α λ,ξ ([12]; x 1 , x 2 ) = ( ξ ⊗ id)((1 ⊗ x 1 ) ⊢ λ (1 ⊗ x 2 )) . W e claim that α λ,ξ ∈ C Y 2 ( D , D ) is a 2-co cycle. This is b ecause δ α λ,ξ ([321]; x 1 , x 2 , x 3 ) = x 1 ⊣ α λ,ξ ([21]; x 2 , x 3 ) − α λ,ξ ([21]; x 1 ⊣ x 2 , x 3 ) + α λ,ξ ([21]; x 1 , x 2 ⊣ x 3 ) − α λ,ξ ([21]; x 1 , x 2 ) ⊣ x 3 = x 1 ⊣ ( ξ ⊗ id )((1 ⊗ x 2 ) ⊢ λ (1 ⊗ x 3 )) − ( ξ ⊗ id)(1 ⊗ ( x 1 ⊣ x 2 ) ⊣ λ 1 ⊗ x 3 ) + ( ξ ⊗ id)(1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ⊣ x 3 ) − ( ξ ⊗ id )(1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) ⊣ x 3 . MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 7 If ǫ den otes the fixed augmen tation of the algebra A , then ǫ ⊗ id : (1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 − 1 ⊗ x 1 ⊣ x 2 ) = 0 , i.e. 1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 − 1 ⊗ x 1 ⊣ x 2 ∈ m ⊗ D . So, ( ξ ⊗ id )((1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) ⊣ λ (1 ⊗ x 3 )) =( ξ ⊗ id )(((1 ⊗ x 1 ⊣ x 2 ) + X i m i ⊗ y i ) ⊣ λ (1 ⊗ x 3 ))) =( ξ ⊗ id )((1 ⊗ x 1 ⊣ x 2 ) ⊣ λ (1 ⊗ x 3 )) + ( ξ ⊗ id )( X i ( m i ⊗ y i ) ⊣ λ (1 ⊗ x 3 )) =( ξ ⊗ id )((1 ⊗ x 1 ⊣ x 2 ) ⊣ λ (1 ⊗ x 3 )) + ( ξ ⊗ id )( X i m i (1 ⊗ y i ) ⊣ λ (1 ⊗ x 3 )) = α λ,ξ ([21]; x 1 ⊣ x 2 , x 3 ) + ( ξ ⊗ id) X i m i (1 ⊗ y i ⊣ λ 1 ⊗ x 3 ) . Note that in the second step from the end w e mak e use of the action of th e alge bra A on A ⊗ D . No w w e hav e 1 ⊗ y i ⊣ λ 1 ⊗ x 3 − 1 ⊗ y i ⊣ x 3 ∈ m ⊗ D , 1 ⊗ y i ⊣ λ 1 ⊗ x 3 = 1 ⊗ y i ⊣ x 3 + h, where h ∈ m ⊗ D . Hence, m i (1 ⊗ y i ⊣ λ 1 ⊗ x 3 ) = m i (1 ⊗ y i ⊣ x 3 + h ) . Since m 2 = 0, we ha v e m i h = 0 . So, m i (1 ⊗ y i ⊣ λ 1 ⊗ x 3 ) = m i ⊗ ( y i ⊣ x 3 ) , making use of the actio n of A on A ⊗ D . Next ( ξ ⊗ id ) X i m i (1 ⊗ y i ⊣ λ 1 ⊗ x 3 ) = X i ( ξ ⊗ id )( m i ⊗ y i ⊣ x 3 ) = X i ξ ( m i )( y i ⊣ x 3 ) = X i ( ξ ( m i ) y i ⊣ x 3 ) = ( ξ ⊗ id )( X i m i ⊗ y i ) ⊣ x 3 = ( ξ ⊗ id ) { (1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) − 1 ⊗ x 1 ⊣ x 2 } ⊣ x 3 = (( ξ ⊗ id)(1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) ⊣ x 3 [using ξ (1) = 0] = α λ,ξ ([12]; x 1 , x 2 ) ⊣ x 3 . Th us, ξ ⊗ id ((1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) ⊣ λ (1 ⊗ x 3 )) = α λ,ξ ([21]; x 1 ⊣ x 2 , x 3 ) + α λ,ξ ([21]; x 1 , x 2 ) ⊣ x 3 . In the same w ay , ξ ⊗ id (1 ⊗ x 1 ⊣ λ (1 ⊗ x 2 ⊣ λ (1 ⊗ x 3 )) = x 1 ⊣ α λ,ξ ([21]; x 2 , x 3 ) + α λ,ξ ([21]; x 1 , x 2 ⊣ x 3 ) . 8 ALICE FIALOWS KI AND ANIT A MAJUM DAR Since ξ ⊗ id((1 ⊗ x 1 ⊣ λ 1 ⊗ x 2 ) ⊣ λ (1 ⊗ x 3 )) − ξ ⊗ id (1 ⊗ x 1 ⊣ λ (1 ⊗ x 2 ⊣ λ 1 ⊗ x 3 )) = 0 , w e ha ve δ α λ,ξ ([321]; x 1 , x 2 , x 3 ) = 0 , and w e can also sho w that δ α λ,ξ ( y ; x 1 , x 2 , x 3 ) = 0 for all y ∈ { [312] , [131] , [213] , [123] } . The f ollo wing p r op osition classifies all infi nitesimal d eformations of D o v er finite di- mensional bases. Prop osition 4.1. F or any infinitesimal deformatio n λ of a dialgebr a D with a finite dimensional b ase A ther e exists a unique homomo rphism φ : K ⊕ H Y 2 ( D , D ) ′ − → A such that λ is e quivalent to the push-out φ ∗ η D . Pro of. Let a λ,ξ ∈ H Y 2 ( D , D ) b e the cohomology class of the co cycle α λ,ξ , corre- sp ond ing to ξ ∈ m ′ . Thus w e ha ve the follo wing homomorphisms: α λ : m ′ − → C Y 2 ( D , D ) a λ : m ′ − → H Y 2 ( D , D ) . Step 1. W e sh o w that the d eformations λ, λ ′ are equ iv alen t if and only if a λ = a λ ′ . Let λ 1 and λ 2 b e tw o equiv alen t d eform ations of th e dialgebra D , w ith b ase A . By definition, there exists a A -linear dialgebra isomorphism ρ : A ⊗ D − → A ⊗ D , such that ( ǫ ⊗ id) ◦ ρ = ǫ ⊗ id . (4.1.1) Since A ⊗ D = D ⊕ ( m ⊗ D ), the isomorphism ρ can b e written as ρ = ρ 1 + ρ 2 where ρ 1 : D − → D and ρ 2 : D − → m ⊗ D . By using equation (4.1.1), w e get ρ 1 = id . Note that by the adjun ction prop ert y of tensor pro d ucts, Hom( D ; m ⊗ D ) ∼ = m ⊗ Hom( D , D ) ∼ = Hom( m ′ ; Hom( D, D )) , where the isomorph isms are giv en b y ρ 2 7− → k X 1 m i ⊗ φ i 7− → k X i χ i . (4.1.2) Here φ i = ( ξ i ⊗ id) ◦ ρ 2 and χ i ( ξ j ) = δ i,j φ i , where { m i } 1 ≤ i ≤ k is a basis of m and { ξ j } 1 ≤ j ≤ k is a basis of m ′ . W e h a ve by equatio n (4.1 .2), ρ (1 ⊗ x ) = ρ 1 (1 ⊗ x ) + ρ 2 (1 ⊗ x ) = 1 ⊗ x + P k 1 m i ⊗ φ i ( x ) . Using the n otation ∗ = {⊣ , ⊢} , the map ρ is a dialgebra homomorphism iff ρ (1 ⊗ x 1 ∗ λ 1 1 ⊗ x 2 ) = ρ (1 ⊗ x 1 ) ∗ λ 2 ρ (1 ⊗ x 2 ) , Let us set ψ r i = α λ r ,ξ i , i = 1 , 2 , . . . , k and r = 1 , 2 . Then w e hav e MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 9 1 ⊗ x 1 ⊣ λ r 1 ⊗ x 2 = 1 ⊗ x 1 ⊣ x 2 + k X i m i ⊗ ψ r i ([21]; x 1 , x 2 ) (4.1.3) and 1 ⊗ x 1 ⊢ λ r 1 ⊗ x 2 = 1 ⊗ x 1 ⊢ x 2 + k X i m i ⊗ ψ r i ([12]; x 1 , x 2 ) . (4.1.4) Therefore, using th e fact that m i .m j = 0 for elemen ts m i , m j ∈ m , ρ (1 ⊗ x 1 ⊣ λ 1 1 ⊗ x 2 ) = 1 ⊗ x 1 ⊣ x 2 + k X i =1 m i ⊗ φ i ( x 1 ⊣ x 2 ) + k X i =1 m i (1 ⊗ ψ 1 i ([21]; x 1 , x 2 )) . Similarly , ρ (1 ⊗ x 1 ⊢ λ 2 1 ⊗ x 2 ) = 1 ⊗ x 1 ⊢ x 2 + k X i =1 m i ⊗ φ i ( x 1 ⊢ x 2 ) + k X i =1 m i (1 ⊗ ψ 1 i ([12]; x 1 , x 2 )) . Again, ρ (1 ⊗ x 1 ) ⊣ λ 2 ρ (1 ⊗ x 2 ) =1 ⊗ ( x 1 ⊣ x 2 ) + k X i =1 m i ⊗ ( ψ 2 i ([21]; x 1 , x 2 )) + k X i =1 m i ⊗  x 1 ⊣ φ i ( x 2 )  + k X i =1 m i ⊗ ( φ i ( x 1 ) ⊣ x 2 ) , and ρ (1 ⊗ x 1 ) ⊢ λ 2 ρ (1 ⊗ x 2 ) =1 ⊗ ( x 1 ⊢ x 2 ) + k X i =1 m i ⊗ ( ψ 2 i ([21]; x 1 , x 2 )) + k X i =1 m i ⊗  x 1 ⊢ φ i ( x 2 )  + k X i =1 m i ⊗ ( φ i ( x 1 ) ⊢ x 2 ) . Th us, the follo wing are equiv alen t: a) ρ (1 ⊗ x 1 ⊣ λ 1 1 ⊗ x 2 ) = ρ (1 ⊗ x 1 ) ⊣ λ 2 ρ (1 ⊗ x 2 ) b) k X i =1 m i ⊗ ( ψ 2 i ([21]; x 1 , x 2 ) − ψ 1 i [21]; x 1 , x 2 )) + k X i =1 m i ⊗ δ φ i ([21]; x 1 , x 2 ) = 0 c) ψ 1 i ([21]; x − 1 , x 2 ) − ψ 2 i ([21]; x 1 , x 2 ) = δ φ i ([21]; x 1 , x 2 ) 10 ALICE FIALOWS KI AND ANIT A MAJUM DAR and similarly these are equiv alen t, to o: a ′ ) ρ (1 ⊗ x 1 ⊢ λ 1 1 ⊗ x 2 ) = ρ (1 ⊗ x 1 ) ⊢ λ 2 ρ (1 ⊗ x 2 ) , b ′ ) k X i =1 m i ⊗ ( ψ 2 i ([12]; x 1 , x 2 ) − ψ 1 i [12]; x 1 , x 2 )) + k X i =1 m i ⊗ δ φ i ([12]; x 1 , x 2 ) = 0 c ′ ) ψ 1 i ([12]; x − 1 , x 2 ) − ψ 2 i ([12]; x 1 , x 2 ) = δ φ i ([12]; x 1 , x 2 ) . Hence, α λ 1 ,ξ i − α λ 2 ,ξ i = δφ i for i ∈ { 1 , 2 , . . . , k } if and only if a λ 1 = a λ 2 . This pro v es step 1. Step 2 . Let φ = id ⊕ a ′ λ : K ⊕ H Y 2 ( D , D ) ′ − → K ⊕ m = A. Claim: φ ∗ η D is equiv alen t to λ . I t follo ws from definitions that α φ ∗ η D = µ ◦ a λ . Thus, a φ ∗ η D = a λ . Hence b y step 1, φ ∗ η D and λ are isomorph ic. This completes the p ro of of Prop osition 4.1.  Let A b e a lo cal algebra with d im ( A/ m 2 ) < ∞ . Th en , A/ m 2 is also lo cal with the maximal ideal m / m 2 , and ( m / m 2 ) 2 = 0 . Definition 4.2. The linear dual space Hom( m / m 2 , K ) is called the tangen t sp ace of A , and is denoted b y T A . Definition 4.3. Let λ b e a d eform ation of D with b ase A . Th en the mapping a π ∗ λ : T A = ( m / m 2 ) ′ − → H Y 2 ( D , D ) , where π is the pro jection A − → A/ m 2 , is called the differential of λ and is denoted b y dλ. Definition 4.4. A formal d eform ation η of a dialgebra D with base B is called miniv ersal if (1) for an y formal deformation λ of a dialgebra D with any local base A ther e exists a h omomorphism f : B → A such that the d eformation λ is equiv alen t to f ∗ η ; (2) with th e ab ov e n otatio ns if A satisfies the condition m 2 = 0, then f is un ique. If η satisfies only condition (1), then it is called v ersal . The follo wing prop osition takes its shap e from the general results of Sc h lessinger [13]. It w as fi rst sho wn for the case of Lie algebras in [2], and stated for Leibniz algebras in [4]. It is straigh tforward to see that it is true for the case of dialgebras, to o. Prop osition 4.5. If the dimension of H Y 2 ( D , D ) is finite, then ther e exists a miniversal deformatio n of the dialgebr a D .  MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 11 5. Some F acts about Har rison Cohomol ogy Let A denote a comm utativ e algebra o v er K . In this section we shall s tate a few results, without pro of [9], ab out Harrison cohomolog y groups of A with co efficients in a A -mo dule M . Let C h ( A ) = { C h q ( A ) , δ } denote th e Harrison complex of A . Definition 5.1. F or an A -mo dule M , the Harrison homology and cohomology of A with co efficien ts in M are defined as follo ws: H H ar r q ( A ; M ) = H q ( C h ( A ) ⊗ M ) , H q H ar r ( A ; M ) = H q (Hom( C h ( A ) , M ); Prop osition 5.2. (1) H 1 H ar r ( A ; M ) is the sp ac e of derivations A → M . (2) Elements of H 2 H ar r ( A ; M ) c orr esp ond b ije ctively to isomo rphism classes of extensions 0 → M → B → A → 0 of the algebr a A by me ans of M .  Corollary 5.3. If A is a lo c al algebr a with the maximal ide al m , then H 1 H ar r ( A ; K ) = ( m / m 2 ) ′ = T A. Prop osition 5.4. Supp ose 0 → M r − i → B r − 1 − p → A → 0 is an r -dimensional extension of A . Then ther e is a ( r − 1) -dimensional extension 0 → M r − 1 i → B r p → A → 0 of A and a 1 -dimensional extension 0 → K i ′ → B r p ′ → B r − 1 → 0 .  Prop osition 5.5. L e t 0 → M i → B p → A → 0 b e an extension of an algebr a A by M . (1) If A has an identity then so do es B . (2) If A is lo c al with the maximal ide al m , then B is lo c al with the maximal ide al p − 1 ( m ) .  Definition 5.6. Tw o extensions B and B ′ of the algebra A by M are said to b e equiv alen t if there exists a K -algebra isomorphism f : B → B ′ suc h that the follo wing diagram comm u tes. 0 − → M i 1 − → B p 1 − → A − → 0 ↓ id ↓ f ↓ id 0 − → M i 2 − → B ′ p 2 − → A − → 0 . An equiv alence from B to B is said to b e an automorph ism of B o v er A . Prop osition 5.7. H 1 H ar r ( A ; M ) is i somorph ic to the set of automorph isms of any given extension 0 → M i → B p → A → 0 of A by M .  12 ALICE FIALOWS KI AND ANIT A MAJUM DAR 6. Obstructions to Exte nding Deforma tions Let A b e a finite dimen s ional commutati v e, u nital, lo cal algebra with a fixed aug- men tation ǫ , and let λ b e a deformation of a dialgebra D with base A . Let 0 → K i → B p → A → 0 b e an extension of A , corresp ond ing to a cohomology class f ∈ H 2 H ar r ( A ; K ). Let q : A → B b e a splitting. Let b ǫ : B → K b e the augmentati on of B . Let I = i ⊗ id : D = K ⊗ D → B ⊗ D and P = p ⊗ id : B ⊗ D → A ⊗ D . Let E = b ǫ ⊗ id : B ⊗ D → K ⊗ D = D and let Q = q ⊗ id : A ⊗ D → B ⊗ D . W e d efine t wo B -bilinear op erations { , } ⊣ , { , } ⊢ on B ⊗ D as follo ws: Let l 1 , l 2 ∈ B ⊗ D . Define { l 1 , l 2 } ⊣ = Q { P ( l 1 ) ⊣ λ P ( l 2 ) } + I [ I − 1 ( l 1 − Q ◦ P ( l 1 )) ⊣ I − 1 ( l 2 − Q ◦ P ( l 2 ))] , { l 1 , l 2 } ⊢ = Q { P ( l 1 ) ⊢ λ P ( l 2 ) } + I [ I − 1 ( l 1 − Q ◦ P ( l 1 )) ⊢ I − 1 ( l 2 − Q ◦ P ( l 2 ))] . It is easy to verify that the t wo op erations thus d efi ned satisfy the follo wing prop erties: ( i ) P { l 1 , l 2 } ∗ = P ( l 1 ) ∗ P ( l 2 ) , where ∗ ∈ {⊣ , ⊢} , l 1 , l 2 ∈ B ⊗ D , (6.0.1) ( ii ) { I ( l ) , l 1 } ∗ = I [ l ∗ E ( l 1 )] , where ∗ ∈ {⊣ , ⊢} , l ∈ D , l 1 ∈ B ⊗ D . (6.0.2) Using the ab o v e t wo prop erties, one can sh o w that E { l 1 , l 2 } ⊣ = E ( l 1 ) ⊣ E ( l 2 ) E { l 1 , l 2 } ⊢ = E ( l 1 ) ⊢ E ( l 2 ) . W e d efine φ ([321] ; l 1 , l 2 , l 3 ) = { l 1 , { l 2 , l 3 } ⊣ } ⊣ − {{ l 1 , l 2 } ⊣ , l 3 } ⊣ , (6.0.3) φ ([312] ; l 1 , l 2 , l 3 ) = {{ l 1 , l 2 } ⊣ , l 3 } ⊣ − { l 1 , { l 2 , l 3 } ⊢ } ⊣ , (6.0.4) φ ([131] ; l 1 , l 2 , l 3 ) = { l 1 , { l 2 , l 3 } ⊣ } ⊣ − {{ l 1 , l 2 } ⊢ , l 3 } ⊣ , (6.0.5) φ ([213] ; l 1 , l 2 , l 3 ) = {{ l 1 , l 2 } ⊣ , l 3 } ⊢ − { l 1 , { l 2 , l 3 } ⊢ } ⊢ , (6.0.6) φ ([213] ; l 1 , l 2 , l 3 ) = { l 1 , { l 2 , l 3 } ⊢ } ⊢ − {{ l 1 , l 2 } ⊢ , l 3 } ⊢ . (6.0.7) It is easy to see that φ ( y ; l 1 , l 2 , l 3 ) ∈ ke r P for all y ∈ Y 3 . Also, note that if an y l i ∈ k er E , i ∈ { 1 , 2 , 3 } , then φ ( l 1 , l 2 , l 3 ) = 0 . Th is defines the map φ : K [ Y 3 ] ⊗ D ⊗ 3 = K [ Y 3 ] ⊗ (( B ⊗ D ) / k er E ) ⊗ 3 → k er P = D . (6.0.8) Th us φ ∈ C Y 3 ( D , D ) . One can c hec k that δ φ = 0 . Let f ′ b e cohomologous to f , and let 0 → K i ′ → B ′ p ′ → A → 0 b e the extension corresp ondin g to f ′ , whic h is isomorphic to the extension corresp ond ing to f . S ince B and B ′ are isomorphic, without loss of generalit y , we shall work with B . Let { , } ′ ∗ , ∗ ∈ {⊣ , ⊢} b e another set of B -bilinear op erations on B ⊗ D , satisfying (1) and (2) ab ov e. Then { l 1 , l 2 } ′ ∗ − { l 1 , l 2 } ∗ ∈ k er P , ∗ ∈ {⊣ , ⊢} for all l 1 , l 2 ∈ B ⊗ D . Also, { l 1 , l 2 } ′ ∗ − { l 1 , l 2 } ∗ = 0 , ∗ ∈ {⊣ , ⊢} if l i ∈ k er E , i ∈ { 1 , 2 } . Th is determines a map ψ : K [ Y 2 ] ⊗ D ⊗ 2 = K [ Y 2 ] ⊗ (( B ⊗ D ) / ker E ) ⊗ 2 → ker P = D. Thus, ψ defines a 2-co c hain. Also, giv en an arbitrary ψ ∈ C Y 2 ( D , D ), there exists an app ropriate { , } ′ ∗ suc h that ψ can b e obtained as { , } ′ ∗ − { , } ∗ , where ∗ ∈ {⊣ , ⊢} . MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 13 W e remark here that if φ , φ ′ ∈ C Y 3 ( D , D ) are the co chains corresp onding to { , } ∗ , { , } ′ ∗ in the sense of the construction ab o v e, then φ ′ − φ = δ ψ . Let O λ ( f ) ∈ H Y 3 ( D , D ) b e the cohomology class of the co c hain φ. W e d efine th e follo wing linear map. O λ : H 2 H ar r ( A, K ) − → H Y 3 ( D , D ) , f 7→ O λ ( f ) . W e thus mak e the f ollo wing prop osition. Prop osition 6.1. The deformation λ with b ase A c an b e extende d to a deformation of the dialgebr a D with b ase B if and only if O λ ( f ) = 0 .  The cohomology class O λ ( f ) is called the obstruction to the extension of the defor- mation λ from A to B . 7. Extendible Deforma tions Let A b e a finite dimen s ional commutati v e, u nital, lo cal algebra with a fixed aug- men tation ǫ , and let λ b e a deformation of a dialgebra D with base A . Let 0 → K i → B p → A → 0 b e an extension of A , corresp ond ing to a cohomology class f ∈ H 2 H ar r ( A ; K ). F ollo wing the same argumen ts as in [3], w e ca n state the follo wing prop osition. Prop osition 7.1. H Y 2 ( D , D ) op er ates tr ansitively on the set of e quivalenc e classes of deforma tions µ of the dialgebr a D with b ase B suc h that p ∗ µ = λ.  W e remark h er e that the group of automorphisms of the extension 0 → K i → B p → A → 0 is H 1 H ar r ( A ; K ) , (5.7) and H 1 H ar r ( A ; K ) = ( m / m 2 ) ′ = T A, (5.3). Note that b y 4.3, there exists a map dλ : T A → H Y 2 ( D , D ) . The group of automorphism s of the extension 0 → K i → B p → A → 0 op erates on th e set of equiv alence classes of deformations µ such that p ∗ µ = λ. W e h a ve the next prop osition, the pro of of which is straigh tforward. Prop osition 7.2. The op er ation of H Y 2 ( D , D ) on the set of e quivalenc e classes of deformat ions µ such that p ∗ µ = λ and the op er ation of the gr oup of automor- phisms of the extension 0 → K i → B p → A → 0 ar e r elat e d by the differ ential dλ : T A → H Y 2 ( D , D ) . In other wor d s, if r : B → B determines an automor- phism of the extension 0 → K i → B p → A → 0 which c orr esp onds to an element h ∈ H 1 H ar r ( A ; K ) = T A, then for any deformation µ of D with b ase B such that p ∗ µ = λ, the differ enc e b etwe en the push-out r ∗ µ and µ is a c o cycle of the c ohom olo gy class dλ ( h ) .  14 ALICE FIALOWS KI AND ANIT A MAJUM DAR Corollary 7.3. Supp ose that the differ ential map dλ : T A − → H Y 2 ( D , D ) is onto. Then the gr oup of automorphisms of the extension 0 → K i → B p → A → 0 op er ates tr ansitively on the set of e quivalenc e classes of deforma tions µ of D with b ase B such that p ∗ µ = λ. The p ro of of the follo wing prop osition is an im itation of the pro of presen ted in [4 ], for Leibniz algebras. Prop osition 7.4. L et A 1 and A 2 b e two finite dimensional lo c al algebr as with aug- mentations ǫ 1 and ǫ 2 , r esp e ctively. L et φ : A 2 − → A 1 b e an algebr a homomor phism with φ (1) = 1 and ǫ 1 ◦ φ = ǫ 2 . Supp ose λ 2 is a deformat ion of a dialgebr a D with b ase A 2 and λ 1 = φ ∗ λ 2 is the push-out via φ . Then the fol lowing diagr am c ommutes: H 2 H ar r ( A 1 ; K ) φ ∗ − → H 2 H ar r ( A 2 ; K ) θ λ 1 ց ւ θ λ 2 H Y 3 ( D ; D ) . Pro of. Let [ f A 1 ] ∈ H 2 H ar r ( A 1 ; K ) corresp ond to the extension 0 − → K i 1 − → A ′ 1 p 1 − → A 1 − → 0 . Also, let [ f A 2 ] = φ ∗ ([ f A 1 ]) ∈ H 2 H ar r ( A 2 ; K ) corresp ond to the extension 0 − → K i 2 − → A ′ 2 p 2 − → A 2 − → 0 . Let q k : A k − → A ′ k b e sections of p k for k = 1 , 2. There exist K -mo dule isomor- phisms A ′ k ∼ = A k ⊕ K . Let ( b, x ) q k denote th e in v erse of ( b, x ) ∈ A k ⊕ K und er the isomorphisms. Define a linear map ψ : A ′ 2 ∼ = ( A 2 ⊕ K ) − → A ′ 1 ∼ = ( A 1 ⊕ K ) b y ψ (( a, x ) q 2 ) = ( φ ( a ) , x ) q 1 for ( a, x ) q 2 ∈ A ′ 2 . Thus we ha v e a morp hism of extensions 0 − → K i 2 − → A ′ 2 p 2 − → A 2 − → 0 ↓ id ↓ ψ ↓ φ 0 − → K i 1 − → A ′ 1 p 1 − → A 1 − → 0 . Let I k = i k ⊗ id , P k = p k ⊗ id an d E k = b ǫ k ⊗ id , wh ere b ǫ k = ǫ k ◦ p k for k = 1 , 2. If m A k denote th e unique maximal ideal of A k then m A ′ k = p − 1 k ( m A k ) is the unique maximal id eal of A ′ k . Let the b asis of m A k and m A ′ k b e { m k i } 1 ≤ i ≤ r k and { n k i } 1 ≤ i ≤ r k +1 resp ectiv ely , for k = 1 , 2. Note that, n k j = ( m k j , 0) q k for 1 ≤ j ≤ r k and n k r k +1 = (0 , 1) q k . The d ialgebra pro ducts on A 2 ⊗ D is giv en by (1 ⊗ x 1 ) ⊣ λ 2 (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊣ x 2 ) + P r 2 i =1 m 2 i ⊗ ψ 2 i ([21]; x 1 , x 2 ) (1 ⊗ x 1 ) ⊢ λ 2 (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊢ x 2 ) + P r 2 i =1 m 2 i ⊗ ψ 2 i ([12]; x 1 , x 2 ) for x 1 , x 2 ∈ D and ψ 2 i = α λ 2 ,ξ 2 i , where { ξ 2 i } is the dual basis of { m 2 i } . MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 15 Let φ ( m 2 i ) = P r 1 j =1 c i,j m 1 j , c i,j ∈ K for 1 ≤ i ≤ r 2 and 1 ≤ j ≤ r 1 . Then the push-out λ 1 = φ ∗ λ 2 on A 1 ⊗ D is defined by (1 ⊗ x 1 ) ⊣ λ 1 (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊣ x 2 ) + P r 2 i =1 ( P r 1 j =1 c i,j m 1 j ) ⊗ ψ 2 i ([21]; x 1 , x 2 ) = 1 ⊗ ( x 1 ⊣ x 2 ) + P r 1 i =1 m 1 j ⊗ ψ 1 j ([21]; x 1 , x 2 ) (1 ⊗ x 1 ) ⊢ λ 1 (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊢ x 2 ) + P r 2 i =1 ( P r 1 j =1 c i,j m 1 j ) ⊗ ψ 2 i ([12]; x 1 , x 2 ) = 1 ⊗ ( x 1 ⊢ x 2 ) + P r 1 i =1 m 1 j ⊗ ψ 1 j ([12]; x 1 , x 2 ) where ψ 1 j ∈ C Y 2 ( D , D ) id defin ed by ψ 1 j ([21]; x 1 , x 2 ) = P r 2 i =1 c i,j ψ 2 i ([21]; x 1 , x 2 ) ψ 1 j ([12]; x 1 , x 2 ) = P r 2 i =1 c i,j ψ 2 i ([12]; x 1 , x 2 ) for x 1 , x 2 ∈ D . F or any 2-co c hain χ ∈ C Y 2 ( D , D ), let us define A ′ k bilinear op erations { , } ⊣ ,k , { , } ⊢ ,k : ( A ′ k ⊗ D ) ⊗ 2 − → A ′ k ⊗ D b y lifting λ k , (1 ⊗ x 1 ) ⊣ k (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊣ x 2 ) + P r k j =1 n k j ⊗ ψ k j ([21]; x 1 , x 2 ) + n k r k +1 χ ([21]; x 1 , x 2 ) (1 ⊗ x 1 ) ⊢ k (1 ⊗ x 2 ) = 1 ⊗ ( x 1 ⊢ x 2 ) + P r k j =1 n k j ⊗ ψ k j ([12]; x 1 , x 2 ) + n k r k +1 χ ([12]; x 1 , x 2 ) for k = 1 , 2 and x 1 , x 2 ∈ D . The op erations { , } ⊣ ,k , { , } ⊢ ,k , for k = 1 , 2 satisfy the conditions (i) an d (ii) of 6.0.1. W e shall sho w that ψ ⊗ id : A ′ 2 ⊗ D − → A ′ 1 ⊗ D p r eserv es the liftings. It is enough to sho w that ( ψ ⊗ id )(1 ⊗ x 1 ∗ 2 1 ⊗ x 2 ) = ψ ⊗ id (1 ⊗ x 1 ) ∗ 1 ψ ⊗ id (1 ⊗ x 2 ) , where ∗ ∈ {⊣ , ⊢} and x 1 , x 2 ∈ D . No w ( ψ ⊗ id )(1 ⊗ x 1 ⊣ 2 1 ⊗ x 2 ) = ψ (1) ⊗ ( x 1 ⊣ x 2 ) + r 2 X j =1 ψ (1) ψ ( n 2 j ) ⊗ ψ 2 j ([21]; x 1 , x 2 ) + ψ (1) ψ ( n 2 r 2 +1 ) ⊗ χ ([21]; x 1 , x 2 ) =1 ⊗ ( x 1 ⊣ x 2 ) + r 2 X j =1  r 1 X i =1 c j,i m 1 i  ⊗ ψ 2 j ([21]; x 1 , x 2 ) + n 1 r 1 +1 ⊗ χ ([21]; x 1 , x 1 ) , where w e used that φ ( m 2 j ) = P r 1 i =1 c j,i m 1 i and ψ ( n 2 r 2 +1 ) = ψ ((0 , 1) q 2 ) = ( φ (0) , 1) q 1 = n 1 r 1 +1 . 16 ALICE FIALOWS KI AND ANIT A MAJUM DAR Simplifying, we conclude that ( ψ ⊗ id )(1 ⊗ x 1 ⊣ 2 1 ⊗ x 2 ) = ψ (1) ⊗ ( x 1 ⊣ x 2 ) + r 1 X i =1 ψ (1) m 1 i ⊗ ψ 1 i ([21]; x 1 , x 2 ) + ψ (1) n 1 r 1 +1 ⊗ χ ([21]; x 1 , x 2 ) = ( ψ (1) ⊗ x 1 ) ⊣ 1 ( ψ (1) ⊗ x 2 ) = ψ ⊗ id (1 ⊗ x 1 ) ⊣ 1 ψ ⊗ id (1 ⊗ x 2 ) . Similarly , w e can sho w that ( ψ ⊗ id )(1 ⊗ x 1 ⊢ 2 1 ⊗ x 2 ) = ψ ⊗ id (1 ⊗ x 1 ) ⊢ 1 ψ ⊗ id (1 ⊗ x 2 ) . Let φ k b e d efined by the op erations { , } ⊣ ,k , { , } ⊢ ,k as h as b een defined in 6.0 .3 and φ k the corresp onding co cycle as in 6.0.8. Since, ψ ( n 2 ( r 2 +1) ) = n 1 ( r 1 +1) , we hav e [ φ 2 ] = [ φ 1 ]. Therefore, θ λ 1 ([ f A 1 ]) = [ φ 1 ] = [ φ 2 ] = θ λ 2 ([ f A 2 ]) = θ λ 2 ◦ φ ∗ ([ f A 1 ]) . Hence, θ λ 1 = θ λ 2 ◦ φ ∗ .  8. Construction of a Miniversal Deforma t ion of a Dialgebra An explicit description of the construction of a v ersal deformation of a Lie alge bra is giv en in [3], and of a Leibniz algebra is giv en in [4]. Here w e sk etc h the construction, for the case of a dialgebra, follo wing the same techniques deve lop ed in [3], [4]. Start with a dialgebra D with dim( H Y 2 ( D , D )) < ∞ . Consider the extension 0 − → H Y 2 ( D , D ) ′ i − → C 1 p − → C 0 − → 0 , where C 0 = K , C 1 = K ⊕ H Y 2 ( D , D ) ′ . Let η 1 denote th e u niv ersal infi n itesimal deformation with base C 1 as describ ed in Section 4. Supp ose for some k ≥ 1, we ha v e constru cted a finite dimensional lo cal alge bra C k , and a deformation η k of D with base C k . Let µ : H 2 Harr ( C k ; K ) − → ( C h 2 ( C k )) ′ b e a h omomorphism mapp ing a cohomology class in to a co cycle repr esen ting the class. The dual map of µ f C k : C h 2 ( C k ) − → H 2 Harr ( C k ; K ) ′ corresp onds to the follo wing extension of C k : 0 − → H 2 Harr ( C k ; K ) ′ i k +1 − → C k +1 p k +1 − → C k − → 0 . The obstruction θ ([ f C k ]) ∈ H 2 Harr ( C k ; K ) ′ ⊗ H Y 3 ( D , D ) yields a map ω k : H 2 Harr ( C k ; K ) − → H Y 3 ( D , D ), by adjunction prop erty of tensor pr o ducts, with th e dual map ω ′ k : H Y 3 ( D , D ) ′ − → H 2 Harr ( C k ; K ) ′ . This induces the f ollo wing extension 0 − → cok er ( ω ′ k ) − → C k +1 /i k +1 ◦ ω ′ k ( H Y 3 ( D , D ) ′ ) − → C k − → 0 . MINIVERSAL DEF ORMA TIONS OF DIALGEBRAS 17 This yields th e extension 0 − → (k er( ω k )) ′ i k +1 − → C k +1 p k +1 − → C k − → 0 where C k +1 = C k +1 /i k +1 ◦ ω ′ k ( H Y 3 ( D , D ) ′ ) and i k +1 , p k +1 are the mappings induced b y i k +1 and p k +1 resp ectiv ely . Along the same lines as in [3 ], [4] w e hav e the follo wing prop osition: Prop osition 8.1. The deform ation η k with b ase C k of a dialgebr a D admits an extension to a deformation with b ase C k +1 which is uniqu e u p to an isomorphism and an automorp hism of the e xtension 0 − → (k er ( ω k )) ′ i k +1 − → C k +1 p k +1 − → C k − → 0 .  This p ro cess giv es rise to a s equence of fi n ite dim en sional lo cal algebras C k and deformations η k of the d ialgebra D with base C k K p 1 ← − C 1 p 2 ← − C 2 p 3 ← − . . . p k ← − C k p k +1 ← − C k +1 . . . suc h that p k +1 ∗ η k +1 = η k . By taking the p ro jectiv e limit w e obtain a formal d efor- mation η of D with base C = ← − − lim k →∞ C k . Let d im ( H Y 2 ( D , D )) = n and K [[ H Y 2 ( D , D ) ′ ]] d enote th e formal p o w er series ring in n v ariables. Also let m den ote the uniqu e maximal ideal in K [[ H Y 2 ( D , D ) ′ ]], consisting of all elemen ts with constan t term zero. W e ha v e the follo win g prop osition, whose pro of can b e found in [4]. Prop osition 8.2. The c omplete lo c al algebr a C = ← − − lim k →∞ C k c an b e describ e d as C ∼ = K [[ H Y 2 ( D , D ) ′ ]] /I , wher e I is an ide al c ontaine d in m 2 .  Along the same lines as in [3], [4], we state th e follo wing theorem, pro of of whic h ob eys the same tec hniqu es as dev elop ed in [3]. Theorem 8.3. L et D b e a dialgebr a with d im( H Y 2 ( D , D )) < ∞ . Then the formal deformation η with b ase C as describ e d ab ove is a miniversal deformation of D .  A cknowledgements W e w ould lik e to th ank Professor Goutam Mukher j ee for his in terest in our w ork and the Indian S tatistical Institute, Calcutta, for hospitalit y . 18 ALICE FIALOWS KI AND ANIT A MAJUM DAR Referen ces [1] F elipe, R. An analo gue to F unct ional Analysis in Dialgebr as , Int. Math. F orum 2 (22) (2007) 1069-1091 . [2] Fialo wski, A. An example of f ormal deformations of Lie algebr as , NA TO Conference on deforma- tion th eory of algebras and applications, 1986, Pro ceedings, Kluw er, D ordrech t (1988), 375-401. [3] Fialo wski, A. and F uc hs, D Construction of Miniversal Deformations of Li e algebr as , J. F unct. Anal. 161 (1999) 76-110. [4] Fialo wski, A., Mandal, A., Mukherjee, G., V ersal Deformations of L eibniz Algebr as (to app ear in J. K -Theory). ArXiv: math/QA0702476. [5] F rabetti, A. Dialgebr a (c o)homolo gy with c o effici ents , in: J. -M. Morel, F. T akens, B. T eissier, eds., D ialgebras and R elated Op erads, Lecture Notes in Mathematics, V ol 1763, Springer, Berlin, 2001, pp. 67-103. [6] A. F rabetti, Dialgebr a homolo gy of asso ci ative al gebr as , C. R. Acad. Sci. Par is 325 (1997) 135-140 [7] Gerstenhab er, M. The Cohomolo gy Structur e of an A sso ciative Ring , Ann. of Math. 78 (1963) 267-288. [8] Ginzburg, V . and Kapranov, M. M. Koszul duality f or op er ads , Duke Math Journal, 76(1) (1994) 203-272. [9] Harrison, D.K., Commutative algebr as and c ohomolo gy , T rans. Amer. Math. Soc. 104 (1962), 191-204. (1958), 450-459. [10] Lod a y , J.-L. Dialgebr as in: J. -M. Morel, F. T ak ens, B. T eissier, eds., Dialgebras and Related Op erads, Lecture Notes in Mathematics, V ol 1763, Springer, Berlin, 2001, p p . 7-66. 269-293. [11] Ma jumdar, A. and Mukherjee, G. Deformation the ory of dialgebr as , K-theory 506 (2002), 1-28. [12] Ma jumdar, A . and Mukherjee, G. Di algebr a c ohomolo gy as a G-algebr a , T ran. Amer. Math. So c. 356(6) (2004), 2443-2457. [13] Schless inger, M., F unctors of A rtin rings , T rans. Amer. Math. So c. 130 (1968), 208-222. +- Alice Fialo wski, Institute of Ma thema tics, E ¨ o tv ¨ o s Lor ´ a nd University, 1117, Budapest, Hungar y., E-mail: fial owsk@cs.elt e.hu Anit a Maj umdar, Dep t. of Ma thema tics, India n Institute of Science, Bangalore-560012, India., E-mail: ani ta@math.iis c.ernet.in